\subsection{An example network}

\begin{figure} [h]
\centering
\includegraphics[width=3in]{./Graphics/paperNetwork.jpg}
\caption{Example network with two terminal-linkage classes}
\label{fig:network-small}
\end{figure}

\begin{figure} [htp]
\includegraphics[width=4.5in]{./Graphics/ConcentrationVsIterationExample.jpg}
\caption{Concentration convergence with iteration}
\label{fig:ConcentrationVsIteration}
\end{figure}

In this section we consider a toy network shown in Figure
\ref{fig:network-small}. For this network the number of complexes $n = 7$, the
number of terminal-linkage classes $l = 2$, and the stoichiometric subspace $S =
\operatorname{span}\{ A+E-C, C-A-D, B-C\}$ has dimension $s = 3$. Therefore,
the deficiency for this network is given by $\delta = 7 - 2 - 3 = 2$ and hence
neither of the deficiency 0-1 theorems can be applied to calculate equilibrium
points. However, intuition suggests that a non-zero steady state will exist
because of the weak reversibility of this network. We apply the fixed point formulation
described in Section 2 and use the numerical algorithm to solve for the
strictly positive steady state. Figure \ref{fig:ConcentrationVsIteration}
illustrates the convergence of the fixed point iterations to the steady state.\\
  
\begin{figure} [h]
  \includegraphics[width=4.5in]{./Graphics/EquilibriumVsTotalMassExample.jpg}
  \caption{Equilibrium dependence on total mass} \label{EquilibriumVsTotalMass}
\end{figure}

Figure \ref{EquilibriumVsTotalMass} illustrates the change in steady state as a
function of the \emph{total mass} in the system. The experiment shows that, as
the total mass is increased, species $A$, $B$ and $C$ adjust linearly to the
additional mass, while species $D$ and $E$ stay at the same levels.

This can be explained analytically by the fact that the vector $\mathbf{1}$ lies in range of $Y^{T}$. 
